Finite Group Databasebeta

← #93027

Subgroups — #93027

Order 512.

240 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#12977C211yesmaximal, characteristic
2#8622C211yesmaximal, characteristic
3#8609C211yesmaximal, characteristic
4C22×M32C411yescharacteristic, complemented
5C2×C8×D4C2211yescharacteristic
6C2×C22⋊C16—21no—
7C2×C22⋊C16C411yescharacteristic, complemented
8#1808—21no—
9#1799—21no—
10#1568—21no—
11#1466—21no—
12#1374—21no—
13#1290C412yescomplemented
14#1289C412yes—
15C23×C8C2×C411yescharacteristic
16C23×C8D411yescharacteristic
17C2×C4×D4D411yescharacteristic
18C2×M32—42no—
19C2×M32—41no—
20C22×C16—21no—
21C22×C16—21no—
22C8×D4—41no—
23C8×D4—41no—
24#432—22no—
25C2×#107—41no—
26C2×C4⋊C8C2×C411yescharacteristic
27C2×#99—41no—
28C2×C22⋊C8C2×C411yescharacteristic
29C2×C22⋊C8Q811yescharacteristic
30C2×C4×C8D411yescharacteristic
31#380—22no—
32C22⋊C16—42no—
33C22⋊C16—41no—
34#342—41no—
35#340—41no—
36#335—41no—
37#334—41no—
38#323—41no—
39#323—41no—
40C22×D4SD1611yescharacteristic
41C23×C4C22⋊C411yescharacteristic
42C23×C4C4⋊C411yescharacteristic
43C22×C8—21no—
44C22×C8—21no—
45C22×C8—21no—
46C22×C8—21no—
47C22×C8—41no—
48C22×C8—41no—
49C22×C8—41no—
50C22×C8—41no—
51C22×C8C4211yescharacteristic
52C22×C8C22⋊C411yescharacteristic
53C22×C8C4⋊C411yescharacteristic
54C22×C8C4⋊C411yescharacteristic
55C4×D4—41no—
56C4×D4—41no—
57C2×C4⋊C4—42no—
58C2×C4⋊C4Q1611yescharacteristic
59C2×C22⋊C4—42no—
60C2×C22⋊C4D811yescharacteristic
61C2×C22⋊C4SD1611yescharacteristic
62C2×C42C22⋊C411yescharacteristic
63M32—42no—
64M32—42no—
65C2×C16—22no—
66C2×C16—41no—
67C2×C16—41no—
68C2×C16—41no—
69C2×C16—41no—
70C2×C16—41no—
71#107—81no—
72#107—81no—
73C4⋊C8—41no—
74C4⋊C8—42no—
75#99—82no—
76C22⋊C8—41no—
77C22⋊C8—41no—
78C22⋊C8—42no—
79#97—42no—
80C4×C8—21no—
81C4×C8—21no—
82#95—42no—
83C24#10211yescharacteristic
84C24#10611yescharacteristic
85C2×D4—21no—
86C2×D4—21no—
87C2×D4—41no—
88C2×D4—41no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—41no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—82no—
98C22×C4#9511yescharacteristic
99C22×C4#10211yescharacteristic
100C22×C4#10311yescharacteristic
101C22×C4#10311yescharacteristic
102C22×C4#10711yescharacteristic
103C2×C8—21no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—42no—
109C2×C8—41no—
110C2×C8—41no—
111C2×C8—41no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—81no—
117C2×C8—82no—
118C2×C8#9911yescharacteristic
119C2×C8#9911yescharacteristic
120C4⋊C4—41no—
121C4⋊C4—82no—
122C4⋊C4—82no—
123C22⋊C4—41no—
124C22⋊C4—41no—
125C22⋊C4—82no—
126C22⋊C4—82no—
127C42—21no—
128C42—21no—
129C16—42no—
130C16—81no—
131C23—21no—
132C23—21no—
133C23—21no—
134C23—21no—
135C23—21no—
136C23—41no—
137C23—41no—
138C23—41no—
139C23—41no—
140C23#34011yescharacteristic
141C23#36111yescharacteristic
142C23#36211yescharacteristic
143D4—42no—
144D4—42no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—81no—
162C2×C4—162no—
163C2×C4—162no—
164C2×C4—162no—
165C2×C4H(3,ℤ/4)11yescharacteristic
166C2×C4#34211yescharacteristic
167C8—21no—
168C8—41no—
169C8—41no—
170C8—41no—
171C8—162no—
172C8AΣL(1,𝔽16)11yescharacteristic
173C8#35211yescharacteristic
174C22—21no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—41no—
183C22—41no—
184C22—41no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22#128911yescharacteristic
189C22#138711yescharacteristic
190C22#138911yescharacteristic
191C4—21no—
192C4—41no—
193C4—41no—
194C4—41no—
195C4—41no—
196C4—322no—
197C4#139011yescharacteristic
198C4#139511yescharacteristic
199C2—21no—
200C2—21no—
201C2—41no—
202C2—41no—
203C2—81no—
204C2#722811yescharacteristic
205C2#723511yescharacteristic
206C2#723611yescharacteristic
subgroupG/H
center, socleC22#1289
commutatorC22×C8C42
Frattini, cosocleC2×C8×D4C22
prime radicalC22×D4SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C8×D4 ▷ C22×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C22×M32C4semidirect
7. C2×C22⋊C16C4semidirect
13. #1290C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#129772
#86222
#86092